Band structure of semimagnetic ${\mathrm{Hg}}_{1-y}{\mathrm{Mn}}_y\mathrm{Te}$ quantum wells

The band structure of semimagnetic ${\mathrm{Hg}}_{1-y}{\mathrm{Mn}}_y\mathrm{Te}∕{\mathrm{Hg}}_{1-x}{\mathrm{Cd}}_x\mathrm{Te}$ type-III quantum wells (QW’s) has been calculated using an eight-band $\mathit{k}\bullet \mathit{p}$ model in an envelope function approach. Details of the band structure calculations are given for the Mn-free case $(y=0)$. A mean-field approach is used to take the influence of the $sp\text{−}d$ exchange interaction on the band structure of QW’s with low Mn concentrations into account. The calculated Landau level fan diagram and the density of states of a ${\mathrm{Hg}}_{0.98}{\mathrm{Mn}}_{0.02}\mathrm{Te}∕{\mathrm{Hg}}_{0.3}{\mathrm{Cd}}_{0.7}\mathrm{Te}$ QW are in good agreement with recent experimental transport observations. The model can be used to interpret the mutual influence of the two-dimensional confinement and the $sp\text{−}d$ exchange interaction on the transport properties of ${\mathrm{Hg}}_{1-y}{\mathrm{Mn}}_y\mathrm{Te}∕{\mathrm{Hg}}_{1-x}{\mathrm{Cd}}_x\mathrm{Te}$ QW’s.

Figure 1

Band structure of $n$-type (a) symmetrical and (b), (d) asymmetrical ${\mathrm{Hg}}_{0.98}{\mathrm{Mn}}_{0.02}\mathrm{Te}∕{\mathrm{Hg}}_{0.3}{\mathrm{Cd}}_{0.7}\mathrm{Te}$ QW’s, as well as (c) an asymmetrical $\mathrm{Hg}\mathrm{Te}∕{\mathrm{Hg}}_{0.3}{\mathrm{Cd}}_{0.7}\mathrm{Te}$ QW, all with $d_W=12.2\mathrm{nm}$ at $T=4.2\mathrm{K}$. $k_{F1}=k_F(H1-)$ and $k_{F2}=k_F(H1+)$. Here the $k_{\Vert }$ vector is $k_{\Vert }(1,0)$; however, the difference between $E\mathbf{\left(}{k}_{\Vert }(1,0)\mathbf{\right)}$ and $E\mathbf{\left(}{k}_{\Vert }(1,1)\mathbf{\right)}$ is less then $6\mathrm{meV}$ at $k_F$ and the $k$ dependence shows qualitatively the same behavior. The $E1$ subband [which corresponds to an interface bound state in the quantum well (Ref. 31)] is shown as dashed lines.

Figure 2

Landau levels of the $E2$, $H1$, and $H2$ subbands for an $n$-type $\mathrm{Hg}\mathrm{Te}∕{\mathrm{Hg}}_{0.3}{\mathrm{Cd}}_{0.7}\mathrm{Te}\left(001\right)$ QW as a function of magnetic field (a) with and (b) without the axial approximation. $d_W=12.2\mathrm{nm}$, $n_{2\mathrm{DEG}}=3.47\times {10}^{12}{\mathrm{cm}}^{-2}$, $n_{\mathrm{DL}}=-0.34n_{2\mathrm{DEG}}$, and $T=4.2\mathrm{K}$. The thick line represents the chemical potential. The Landau levels are labeled with quantum numbers $N$, and the arrows (↑,↓) indicate the dominant spin orientation of the state.

Figure 3

Landau levels of the $E2$, $H1$, and $H2$ subbands for an $n$-type ${\mathrm{Hg}}_{0.98}{\mathrm{Mn}}_{0.02}\mathrm{Te}∕{\mathrm{Hg}}_{0.3}{\mathrm{Cd}}_{0.7}\mathrm{Te}\left(001\right)$ QW as a function of magnetic field ($d_W=12.2\mathrm{nm}$, $n_{2\mathrm{DEG}}=3.47\times {10}^{12}{\mathrm{cm}}^{-2}$, $n_{\mathrm{DL}}=-0.34n_{2\mathrm{DEG}}$, and $T=4.2\mathrm{K}$). The thick line represents the chemical potential. As in Fig. 2, the Landau levels are labeled with quantum numbers $N$, and the arrows (↑,↓) indicate the dominant spin orientation of the state.

Figure 4

Density of states of the $n$-type ${\mathrm{Hg}}_{0.98}{\mathrm{Mn}}_{0.02}\mathrm{Te}∕{\mathrm{Hg}}_{0.3}{\mathrm{Cd}}_{0.7}\mathrm{Te}$ QW at the Fermi level (dotted lines) compared with experimental SdH oscillations (solid lines).